The Topology of the External Activity Complex of a Matroid

We prove that the external activity complex Act<(M) of a matroid is shellable. In fact, we show that every linear extension of LasVergnas’s external/internal order <ext/int on M provides a shelling of Act<(M). We also show that every linear extension of LasVergnas’s internal order <int on M provides a shelling of the independence complex IN(M). As a corollary, Act<(M) and M have the same hvector. We prove that, after removing its cone points, the external activity complex is contractible if M contains U1,3 as a minor, and a sphere otherwise.